Chapter 5.5, Problem 1E

a.

To determine

To indicate whether the Axiom of choice must be employed to select one element from each set in the following collections.

Explanation of Solution

Given:

Consider the infinite collection of sets, each set containing one odd and one even integer.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if $\mathrm{{\rm A}}$ is any collection of nonempty sets, then there exists a function $\mathrm{F}$ from $\mathrm{{\rm A}}$ to ${\displaystyle \underset{\mathrm{A}\in \mathrm{{\rm A}}}{\cup }\mathrm{A}}$ such that for every $\mathrm{A}\in \mathrm{{\rm A}}$ , $\mathrm{F}(\mathrm{A})\in \mathrm{A}$

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

b.

To determine

To indicate whether the Axiom of Choice must be employed.

Explanation of Solution

Given:

Consider the finite collection of sets such that each set is uncountable

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if $\mathrm{{\rm A}}$ is any collection of nonempty sets, then there exists a function $\mathrm{F}$ from $\mathrm{{\rm A}}$ to ${\displaystyle \underset{\mathrm{A}\in \mathrm{{\rm A}}}{\cup }\mathrm{A}}$ such that for every $\mathrm{A}\in \mathrm{{\rm A}}$ , $\mathrm{F}(\mathrm{A})\in \mathrm{A}$

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

c.

To determine

To indicate whether the Axiom of Choice must be employed.

Explanation of Solution

Given:

Consider the infinite collection of sets each containing exactly four natural numbers.

Calculation:

Since, in the infinite collection of the sets, each set containing one odd and one even integer that is each set in the infinite collection is nonempty and finite

Then, the collection is infinite collection of nonempty sets but its sets are finite

Since, the Axiom of Choice is that if $\mathrm{{\rm A}}$ is any collection of nonempty sets, then there exists a function $\mathrm{F}$ from $\mathrm{{\rm A}}$ to ${\displaystyle \underset{\mathrm{A}\in \mathrm{{\rm A}}}{\cup }\mathrm{A}}$ such that for every $\mathrm{A}\in \mathrm{{\rm A}}$ , $\mathrm{F}(\mathrm{A})\in \mathrm{A}$

Therefore, there is no need the Axiom of Choice to select one element from each set in the collection.

Hence, there is no need to employ the Axiom of Choice.

d.

To determine

To indicate whether the Axiom of Choice must be employed.

Explanation of Solution

Given:

Consider the denumerable collection of uncountable sets.

Calculation:

Since, the collection is denumerable collection of uncountable sets.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if $\mathrm{{\rm A}}$ is any collection of nonempty sets, then there exists a function $\mathrm{F}$ from $\mathrm{{\rm A}}$ to ${\displaystyle \underset{\mathrm{A}\in \mathrm{{\rm A}}}{\cup }\mathrm{A}}$ such that for every $\mathrm{A}\in \mathrm{{\rm A}}$ , $\mathrm{F}(\mathrm{A})\in \mathrm{A}$

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

e.

To determine

To indicate whether the Axiom of Choice must be employed.

Explanation of Solution

Given:

Consider the collection of following sets-

  $\{\mathrm{A}:\mathrm{\pi }\mathrm{\mathbb{N}}\subseteq \mathrm{A}\subseteq \mathrm{ℝ}-\mathrm{\mathbb{Q}}\}$

Where $\mathrm{\pi }\mathrm{\mathbb{N}}=\{\mathrm{\pi }\mathrm{n}:\mathrm{n}\in \mathrm{\mathbb{N}}\}$

Calculation:

Since the set of natural numbers are infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if $\mathrm{{\rm A}}$ is any collection of nonempty sets, then there exists a function $\mathrm{F}$ from $\mathrm{{\rm A}}$ to ${\displaystyle \underset{\mathrm{A}\in \mathrm{{\rm A}}}{\cup }\mathrm{A}}$ such that for every $\mathrm{A}\in \mathrm{{\rm A}}$ , $\mathrm{F}(\mathrm{A})\in \mathrm{A}$

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

f.

To determine

To indicate whether the Axiom of Choice must be employed.

Explanation of Solution

Given:

Consider the following collection of sets

  $\{\mathrm{A}:\mathrm{A}\subseteq \mathrm{\mathbb{N}}\text{ and both A and }\mathrm{\mathbb{N}}-\mathrm{A}\text{ are infinite}}$

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if $\mathrm{{\rm A}}$ is any collection of nonempty sets, then there exists a function $\mathrm{F}$ from $\mathrm{{\rm A}}$ to ${\displaystyle \underset{\mathrm{A}\in \mathrm{{\rm A}}}{\cup }\mathrm{A}}$ such that for every $\mathrm{A}\in \mathrm{{\rm A}}$ , $\mathrm{F}(\mathrm{A})\in \mathrm{A}$

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

g.

To determine

To indicate whether the Axiom of Choice must be employed.

Explanation of Solution

Given:

Consider the following collection of sets

  $\{\mathrm{A}:\mathrm{A}\subseteq \mathrm{ℝ}\text{ and both A and }\mathrm{ℝ}-\mathrm{A}\text{ are infinite}}$

Calculation:

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if $\mathrm{{\rm A}}$ is any collection of nonempty sets, then there exists a function $\mathrm{F}$ from $\mathrm{{\rm A}}$ to ${\displaystyle \underset{\mathrm{A}\in \mathrm{{\rm A}}}{\cup }\mathrm{A}}$ such that for every $\mathrm{A}\in \mathrm{{\rm A}}$ , $\mathrm{F}(\mathrm{A})\in \mathrm{A}$

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.

h.

To determine

To indicate whether the Axiom of Choice must be employed.

Explanation of Solution

Given:

Consider the following collection of sets

  $\{\mathrm{A}:\mathrm{A}\subseteq \mathrm{ℝ}\text{ and A is denumerable}}$

Calculation:

Since, in the collection of sets, each set in A is infinite.

Then, the collection of sets is infinite and sets are also infinite.

Since, the Axiom of Choice is that if $\mathrm{{\rm A}}$ is any collection of nonempty sets, then there exists a function $\mathrm{F}$ from $\mathrm{{\rm A}}$ to ${\displaystyle \underset{\mathrm{A}\in \mathrm{{\rm A}}}{\cup }\mathrm{A}}$ such that for every $\mathrm{A}\in \mathrm{{\rm A}}$ , $\mathrm{F}(\mathrm{A})\in \mathrm{A}$

Therefore, there is need of the Axiom of Choice to select one element from each set in the collection.

Hence, it is necessary to employ the Axiom of Choice.